HIDALGO, Texas—When 10 Central Americans illegally crossed the Rio Grande into Texas recently, they immediately went looking for the U.S. Border Patrol, hoping their tales of fleeing gang violence back home would earn them asylum.

“This is your daughter?” an agent asked José Alfaro, 42, a street vendor from El Salvador, who said he came to the U.S. with his 3-year-old girl after gang members tried to kill him for the third time.

“Yes,”...

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